Summary: We perform a systematic study of the maximum Lyapunov exponent values \(\lambda\) for the motion of classical closed strings in Anti-de Sitter black hole geometries with spherical, planar and hyperbolic horizons. Analytical estimates from the linearized variational equations together with numerical integrations predict the bulk Lyapunov exponent value as \(\lambda \approx 2 \pi Tn\), where \(n\) is the winding number of the string. The celebrated bound on chaos stating that \(\lambda \leq 2 \pi T\) is thus systematically modified for winding strings in the bulk. Within gauge/string duality, such strings apparently correspond to complicated operators which either do not move on Regge trajectories, or move on subleading trajectories with an unusual slope. Depending on the energy scale, the out-of-time-ordered correlation functions of these operators may still obey the bound \(2 \pi T\), or they may violate it like the bulk exponent. We do not know exactly why the bound on chaos can be modified but the indication from the gauge/string dual viewpoint is that the correlation functions of the dual gauge operators never factorize and thus the original derivation of the bound on chaos does not apply.